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September  2021, 41(9): 4255-4281. doi: 10.3934/dcds.2021035

## Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues

 1 Ingenium College of Liberal Arts, Kwangwoon University, Seoul 01891, Korea 2 Division of Medical Mathematics, National Institute for Mathematical Sciences, Daejeon 34047, Korea 3 Department of Mathematics, Kyung Hee University, Seoul 02447, Korea 4 School of Mathematics, Korea Institute for Advanced Study, Seoul 02455, Korea 5 Samsung Fire & Marine Insurance, Seoul 04523, Korea 6 Department of Undergraduate Studies, Daegu Gyeongbuk Institute of Science and Technology, Daegu 42988, Korea

* Corresponding author: Jongmin Han

Received  September 2020 Published  February 2021

In this paper, we study the dynamic phase transition for one dimensional Brusselator model. By the linear stability analysis, we define two critical numbers ${\lambda}_0$ and ${\lambda}_1$ for the control parameter ${\lambda}$ in the equation. Motivated by [9], we assume that ${\lambda}_0< {\lambda}_1$ and the linearized operator at the trivial solution has multiple critical eigenvalues $\beta_N^+$ and $\beta_{N+1}^+$. Then, we show that as ${\lambda}$ passes through ${\lambda}_0$, the trivial solution bifurcates to an $S^1$-attractor ${\mathcal A}_N$. We verify that ${\mathcal A}_N$ consists of eight steady state solutions and orbits connecting them. We compute the leading coefficients of each steady state solution via the center manifold analysis. We also give numerical results to explain the main theorem.

Citation: Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4255-4281. doi: 10.3934/dcds.2021035
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##### References:
 [1] R. Anguelov and S. M. Stoltz, Stationary and oscillatory patterns in a coupled Brusselator model, Math. Computers Simul., 133 (2017), 39-46.  doi: 10.1016/j.matcom.2015.06.002.  Google Scholar [2] K. J. Brown and F. A. Davidson, Global bifurcation in the Brusselator system, Nonlin. Anal., 12 (1995), 1713-1725.  doi: 10.1016/0362-546X(94)00218-7.  Google Scholar [3] I. R. Epstein and J. A. Pojman, An Introduction to Nonlinear Chemical Dynamics,, Oxford Univ. Press, 1998.   Google Scholar [4] M. Ghergu, Non-constant steady-state solutions for Brusselator type systems, Nonlinearity, 21 (2008), 2331-2345.  doi: 10.1088/0951-7715/21/10/007.  Google Scholar [5] B. Guo and Y. Han, Attractor and spatial chaos for the Brusselator in $\mathbb{R}^N$, Nonlin. Anal., 70 (2009), 3917-3931.  doi: 10.1016/j.na.2008.08.002.  Google Scholar [6] H. Kang and Y. Pesin, Dynamics of a discrete brusselator model: Escape to infinity and julia set, Milan J. Math., 73 (2005), 1-17.  doi: 10.1007/s00032-005-0036-y.  Google Scholar [7] H. Shoji, K. Yamada, D. Ueyama and T. Ohta, Turing patterns in three dimensions, Phys. Rev. E, 75 (2007), 046212, 13 pp. doi: 10.1103/PhysRevE.75.046212.  Google Scholar [8] T. Ma and S. Wang, Bifurcation Theory and Applications, World Scientific, 2005. doi: 10.1142/9789812701152.  Google Scholar [9] T. Ma and S. Wang, Phase transitions for the Brusselator model, J. Math. Phys., 52 (2011), 033501, 23 pp. doi: 10.1063/1.3559120.  Google Scholar [10] T. Ma and S. Wang, Phase Transition Dynamics 2nd ed., Springer, 2019. doi: 10.1007/978-3-030-29260-7.  Google Scholar [11] MathWorks, Matlab: Mathematics(R2020a), Retrieved from https://www.mathworks.com/help/pdf_doc/matlab_math.pdf Google Scholar [12] A. S. Mikhailov and K. Showalter, Control of waves, patterns and turbulence in chemical systems, Physics Reports, 425 (2006), 79-194.  doi: 10.1016/j.physrep.2005.11.003.  Google Scholar [13] L. A. Peletier and W. C. Troy, Spatial Patterns: Higher Order Models in Physics and Mechanics, Birkhauser, 2001. doi: 10.1007/978-1-4612-0135-9.  Google Scholar [14] R. Peng and M. X. Wang, Pattern formation in the Brusselator system, J. Math. Anal. Appl., 309 (2005), 151-166.  doi: 10.1016/j.jmaa.2004.12.026.  Google Scholar [15] R. Peng and M. X. Wang, On steady-state solutions of the Brusselator-type system, Nonlin. Anal., 71 (2009), 1389-1394.  doi: 10.1016/j.na.2008.12.003.  Google Scholar [16] I. Prigogine and R. Lefever, Symmetry breaking instabilities in dissipative systems, J. Chem. Phys., 48 (1968), 1695-1700.   Google Scholar [17] A. Toth, V. Gaspar and K. Showalter, Signal transmission in chemical systems: Propagation of chemical waves through capillary tubes, J. Phys. Chem., 98 (1994), 522-531.  doi: 10.1021/j100053a029.  Google Scholar [18] A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.  Google Scholar [19] Y. You, Global Dynamics of the Brusselator equations, Dynamics of PDE, 4 (2007), 167-196.  doi: 10.4310/DPDE.2007.v4.n2.a4.  Google Scholar
Examples of Structure of ${\mathcal A}_N$ in Table 1, 2 and 3
Examples of Structure of ${\mathcal A}_N$ in Table 4
Case (ⅰ) of (4.3) and $N = 4$. With $w (x,0) = w_0(x)$, (a) $u_h(x,t) \to u_1^+(x)$ and (b) $v_h(x,t) \to v_1^+(x)$. With $w (x,0) = w_1(x)$, (c) $u_h(x,t) \to u_1^+(x)$ and (d) $v_h(x,t) \to v_1^+(x)$
Case (ⅱ) of (4.3) and $N = 4$. With $w (x,0) = w_0(x)$, (a) $u_h(x,t) \to u_2^+(x)$ and (b) $v_h(x,t) \to v_2^+(x)$. With $w (x,0) = w_1(x)$, (c) $u_h(x,t) \to u_1^+(x)$ and (d) $v_h(x,t) \to v_1^+(x)$
Case (ⅲ) of (4.3) and $N = 4$. With $w (x,0) = w_0(x)$, (a) $u_h(x,t) \to u_1^+(x)$ and (b) $v_h(x,t) \to v_1^+(x)$. With $w (x,0) = w_1(x)$, (c) $u_h(x,t) \to u_1^+(x)$ and (d) $v_h(x,t) \to v_1^+(x)$
Case (ⅰ) of (4.3) and $N = 8$. With $w (x,0) = w_0(x)$, (a) $u_h(x,t) \to u_1^-(x)$ and (b) $v_h(x,t) \to v_1^-(x)$. With $w (x,0) = w_1(x)$, (c) $u_h(x,t) \to u_1^+(x)$ and (d) $v_h(x,t) \to v_1^+(x)$
Case (ⅱ) of (4.3) and $N = 8$. With $w (x,0) = w_0(x)$, (a) $u_h(x,t) \to u_1^+(x)$ and (b) $v_h(x,t) \to v_2^+(x)$. With $w (x,0) = w_2(x)$, (c) $u_h(x,t) \to u_1^-(x)$ and (d) $v_h(x,t) \to v_1^-(x)$
Case (ⅲ) of (4.3) and $N = 8$. With $w (x,0) = w_0(x)$, (a) $u_h(x,t) \to u_2^+(x)$ and (b) $v_h(x,t) \to v_2^+(x)$. With $w (x,0) = w_1(x)$, (c) $u_h(x,t) \to u_1^-(x)$ and (d) $v_h(x,t) \to v_1^-(x)$
Stability for $k = 2$
 subcases $w_1^+$ $w_1^-$ $w_2^+$ $w_2^-$ (ⅰ-1) stable saddle $\times$ $\times$ (ⅰ-2) saddle stable $\times$ $\times$ (ⅰ-3) $\times$ $\times$ stable saddle (ⅰ-4) $\times$ $\times$ saddle stable
 subcases $w_1^+$ $w_1^-$ $w_2^+$ $w_2^-$ (ⅰ-1) stable saddle $\times$ $\times$ (ⅰ-2) saddle stable $\times$ $\times$ (ⅰ-3) $\times$ $\times$ stable saddle (ⅰ-4) $\times$ $\times$ saddle stable
Stability for $k = 4$
 subcases $w_1^\pm$ $w_2^\pm$ $w_3^\pm$ $w_4^\pm$ (ⅱ-1) stable saddle $\times$ $\times$ (ⅱ-2) saddle stable $\times$ $\times$ (ⅱ-3) $\times$ $\times$ stable saddle (ⅱ-4) $\times$ $\times$ saddle stable
 subcases $w_1^\pm$ $w_2^\pm$ $w_3^\pm$ $w_4^\pm$ (ⅱ-1) stable saddle $\times$ $\times$ (ⅱ-2) saddle stable $\times$ $\times$ (ⅱ-3) $\times$ $\times$ stable saddle (ⅱ-4) $\times$ $\times$ saddle stable
Stability for $k = 6$
 subcases $w_1^+$ $w_1^-$ $w_2^+$ $w_2^-$ $w_3^\pm$ $w_4^\pm$ (ⅲ-1) stable saddle $\times$ $\times$ saddle stable (ⅲ-2) saddle stable $\times$ $\times$ stable saddle (ⅲ-3) $\times$ $\times$ stable saddle saddle stable (ⅲ-4) $\times$ $\times$ saddle stable stable saddle
 subcases $w_1^+$ $w_1^-$ $w_2^+$ $w_2^-$ $w_3^\pm$ $w_4^\pm$ (ⅲ-1) stable saddle $\times$ $\times$ saddle stable (ⅲ-2) saddle stable $\times$ $\times$ stable saddle (ⅲ-3) $\times$ $\times$ stable saddle saddle stable (ⅲ-4) $\times$ $\times$ saddle stable stable saddle
Stability for $k = 8$
 subcases $w_1^\pm$ $w_2^\pm$ $w_3^\pm$ $w_4^\pm$ (ⅳ-1) stable stable saddle saddle (ⅳ-2) saddle saddle stable stable
 subcases $w_1^\pm$ $w_2^\pm$ $w_3^\pm$ $w_4^\pm$ (ⅳ-1) stable stable saddle saddle (ⅳ-2) saddle saddle stable stable
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